EDUC 200C

Friday, October 26, 2012

Goals for today

• Homework• Midterm exam • Null Hypothesis• Sampling distributions• Hypothesis testing• Mid-quarter evaluations

The null hypothesis

• Example: A study compares the results of a new reading program for middle school students. In this study, 36 students received the experimental reading program.

• Each student’s reading score was measured before and after the program.

• What is the variable of interest?--The variable of interest is the score change for each

student.--Score change is positive if a student’s score improved

and negative if the score got worse and zero if the student’s score didn’t change.

• What is our null hypothesis?

Hypothesis testing vocabulary

Null Hypothesis: A hypothesis to be tested. • Use the symbol H0 (e.g. H0 : X=0)

Alternative Hypothesis: A hypothesis that represents the opposite of the null hypothesis.

• One or the other must be true, there can be no third option

• Use the symbol HA or H1 (e.g. HA : X≠0)

Hypothesis Test: The test of whether the null hypothesis (H0) should be rejected in favor of the alternative hypothesis.

The importance of sample size

and

Sampling Distributions

• Population distribution– Based on all members on a population– μ and σ – Rarely do we know true values, hope to estimate them

• Sample distribution– Based on a single sample of the population– – Calculate these to try to describe the population

• Sampling distribution– From multiple samples, each with own mean and sd– Distribution captures uncertainty about how well the

sample statistics represent the population parameters

-4 -2 0 2 4Sample means

-4 -2 0 2 4Sample means

-4 -2 0 2 4Sample means

-4 -2 0 2 4Sample means

20 samples randomly drawn from a population with mean = 0 and sd = 2

N = 5 N = 10

N = 20 N = 50

Number of samples and the Central Limit Theorem

0.2

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-.5 0 .5mn

0.5

11.

52

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-.6 -.4 -.2 0 .2 .4mn

0.2

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-1.5 -1 -.5 0 .5 1mn

0.5

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-1 -.5 0 .5 1mn

As the number of samples increases, the distribution of sample means approaches a normal distribution with a mean equal to the population mean

10 Samples 20 Samples

50 Samples 100 Samples

Standard error of the mean(different from standard error of the estimate)

• A description of the standard deviation of the distribution of sample means.

• As we just saw, this is related to the sample size.– Larger samples mean smaller variation

in sample means.

How it comes together

• Recalling the reading program example, we want to know whether the post-treatment sample of scores “looks like” the original sample.

• We know, by the CLT, sample means are normally distributed.

• We know how to calculate the standard error of those sample means.

• This allows for the conversion of our sample mean to a z-score that can be compared to the standard normal distribution.

Reading program example, continued

• Pre-treatment scores have a mean of 45 and a standard deviation of 10

• Post-treatment score mean = 49

• H0: post treatment scores = 45 (no change)

• H1: post treatment scores ≠ 45

Reading program example, continued

Post-treatment meanPre-treatment mean(our null hypothesis)

Standard error of meanPopulation standarddeviation

Sample size

Reading program example, continued

Post-treatment meanPre-treatment mean(our null hypothesis)

Standard error of meanPopulation standarddeviation

Sample size

Note: We assume here that our pre-treatmentsample mean and sd are the same as the population parameters.This isn’t quite right, but we discuss how to deal with this next week.

Reject the null

• z = 2.4• Checking pg. 467 in our book we see that a average

score of 49 is greater than 99.18% of the means we would expect to see if the sample had not changed

• Conversely, there is a 0.82 % chance we would see a score of 49 or higher if scores had not changed

• Further, we know 1.64% of sample means will have z-scores as or more extreme (on either side of the mean)

• This exceeds the typical α=.05 threshold for rejecting the null hypothesis

• Thus we reject our null hypothesis with the recognition that there is a 1.64% chance we made a type I error (rejected the null when in fact it was true)

Hypothesis testing vocabulary, part 2

• We never “accept” the null, we just “reject the null” or “fail to reject the null”.– If the null hypothesis is rejected, we

conclude that the alternative hypothesis is true within the scope of our alpha.

– If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alternative hypothesis (i.e. we don’t know at this time).

Mid-quarter section evaluation

Please fill out an evaluation to let me know how to improve this class for you!